Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
41.1-a1 |
41.1-a |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{20} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
|
\( 2^{2} \cdot 5 \) |
$1$ |
$28.02091986$ |
4.00367 |
\( -\frac{357571850055303381213985}{13422659310152401} a^{4} + \frac{540487576413107354164635}{13422659310152401} a^{3} + \frac{1247371673863409551905290}{13422659310152401} a^{2} - \frac{1263890879184018681279920}{13422659310152401} a - \frac{115913951592431810832436}{13422659310152401} \) |
\( \bigl[a^{5} - a^{4} - 3 a^{3} + 2 a^{2} - a\) , \( a^{5} - a^{4} - 3 a^{3} + 2 a^{2} - 3 a\) , \( a^{5} - a^{4} - 3 a^{3} + 2 a^{2} - a\) , \( 607 a^{5} - 963 a^{4} - 3035 a^{3} + 3862 a^{2} + 3594 a - 4032\) , \( -9346 a^{5} + 11784 a^{4} + 60096 a^{3} - 63006 a^{2} - 89033 a + 85118\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-3a^{3}+2a^{2}-a\right){x}{y}+\left(a^{5}-a^{4}-3a^{3}+2a^{2}-a\right){y}={x}^{3}+\left(a^{5}-a^{4}-3a^{3}+2a^{2}-3a\right){x}^{2}+\left(607a^{5}-963a^{4}-3035a^{3}+3862a^{2}+3594a-4032\right){x}-9346a^{5}+11784a^{4}+60096a^{3}-63006a^{2}-89033a+85118$ |
41.1-a2 |
41.1-a |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{2} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
|
\( 2 \) |
$1$ |
$2802.091986$ |
4.00367 |
\( -\frac{734681}{41} a^{4} - \frac{233799}{41} a^{3} + \frac{3907204}{41} a^{2} + \frac{1436078}{41} a - \frac{3784992}{41} \) |
\( \bigl[a^{4} - a^{3} - 3 a^{2} + a + 1\) , \( a^{4} - a^{3} - 3 a^{2} + 2 a + 1\) , \( a^{2} - a - 1\) , \( a^{5} + 4 a^{4} - 9 a^{3} - 12 a^{2} + 11 a - 4\) , \( 2 a^{5} + 5 a^{4} - 18 a^{3} - 15 a^{2} + 28 a - 5\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-3a^{2}+a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{4}-a^{3}-3a^{2}+2a+1\right){x}^{2}+\left(a^{5}+4a^{4}-9a^{3}-12a^{2}+11a-4\right){x}+2a^{5}+5a^{4}-18a^{3}-15a^{2}+28a-5$ |
41.1-a3 |
41.1-a |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{4} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
|
\( 2^{2} \) |
$1$ |
$700.5229965$ |
4.00367 |
\( \frac{2962060985575}{1681} a^{4} + \frac{731644682050}{1681} a^{3} - \frac{15541949609925}{1681} a^{2} - \frac{5156995031725}{1681} a + \frac{14955417009784}{1681} \) |
\( \bigl[a^{5} - a^{4} - 3 a^{3} + 2 a^{2} - a\) , \( a^{5} - a^{4} - 3 a^{3} + 2 a^{2} - 3 a\) , \( a^{5} - a^{4} - 3 a^{3} + 2 a^{2} - a\) , \( 117 a^{5} - 193 a^{4} - 665 a^{3} + 952 a^{2} + 959 a - 1097\) , \( 1324 a^{5} - 2576 a^{4} - 7659 a^{3} + 12919 a^{2} + 11717 a - 14637\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-3a^{3}+2a^{2}-a\right){x}{y}+\left(a^{5}-a^{4}-3a^{3}+2a^{2}-a\right){y}={x}^{3}+\left(a^{5}-a^{4}-3a^{3}+2a^{2}-3a\right){x}^{2}+\left(117a^{5}-193a^{4}-665a^{3}+952a^{2}+959a-1097\right){x}+1324a^{5}-2576a^{4}-7659a^{3}+12919a^{2}+11717a-14637$ |
41.1-a4 |
41.1-a |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{10} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
|
\( 2 \cdot 5 \) |
$1$ |
$112.0836794$ |
4.00367 |
\( \frac{815501597212588028076}{115856201} a^{4} - \frac{2284984698342134580907}{115856201} a^{3} - \frac{1792523287720805559473}{115856201} a^{2} + \frac{6039452497813815714645}{115856201} a + \frac{524452446825637320235}{115856201} \) |
\( \bigl[a^{4} - 4 a^{2} - a + 1\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 3\) , \( a^{5} - a^{4} - 3 a^{3} + 2 a^{2} - 2 a\) , \( 92 a^{5} - 24 a^{4} - 448 a^{3} - 78 a^{2} + 360 a + 40\) , \( 204 a^{5} + a^{4} - 584 a^{3} + 13 a^{2} + 285 a + 44\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}-a+1\right){x}{y}+\left(a^{5}-a^{4}-3a^{3}+2a^{2}-2a\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+6a+3\right){x}^{2}+\left(92a^{5}-24a^{4}-448a^{3}-78a^{2}+360a+40\right){x}+204a^{5}+a^{4}-584a^{3}+13a^{2}+285a+44$ |
41.1-b1 |
41.1-b |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{20} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2^{2} \cdot 5 \) |
$2.280632233$ |
$1.843695156$ |
2.78770 |
\( -\frac{357571850055303381213985}{13422659310152401} a^{4} + \frac{540487576413107354164635}{13422659310152401} a^{3} + \frac{1247371673863409551905290}{13422659310152401} a^{2} - \frac{1263890879184018681279920}{13422659310152401} a - \frac{115913951592431810832436}{13422659310152401} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 3 a\) , \( 1\) , \( -10 a^{4} + 109 a^{3} - 59 a^{2} - 317 a - 71\) , \( -216 a^{4} + 1168 a^{3} - 88 a^{2} - 3288 a - 374\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-10a^{4}+109a^{3}-59a^{2}-317a-71\right){x}-216a^{4}+1168a^{3}-88a^{2}-3288a-374$ |
41.1-b2 |
41.1-b |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{2} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$0.228063223$ |
$115230.9472$ |
2.78770 |
\( -\frac{734681}{41} a^{4} - \frac{233799}{41} a^{3} + \frac{3907204}{41} a^{2} + \frac{1436078}{41} a - \frac{3784992}{41} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 3 a\) , \( 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-a^{3}+a^{2}+3a-1\right){x}$ |
41.1-b3 |
41.1-b |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{4} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{2} \) |
$0.456126446$ |
$28807.73681$ |
2.78770 |
\( \frac{2962060985575}{1681} a^{4} + \frac{731644682050}{1681} a^{3} - \frac{15541949609925}{1681} a^{2} - \frac{5156995031725}{1681} a + \frac{14955417009784}{1681} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 3 a\) , \( 1\) , \( -6 a^{3} + 6 a^{2} + 18 a - 11\) , \( 4 a^{4} - 2 a^{3} - 18 a^{2} + 2 a + 16\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-6a^{3}+6a^{2}+18a-11\right){x}+4a^{4}-2a^{3}-18a^{2}+2a+16$ |
41.1-b4 |
41.1-b |
$4$ |
$10$ |
6.6.1279733.1 |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( 41^{10} \) |
$137.75168$ |
$(a^5-2a^4-3a^3+6a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2 \cdot 5 \) |
$1.140316116$ |
$7.374780625$ |
2.78770 |
\( \frac{815501597212588028076}{115856201} a^{4} - \frac{2284984698342134580907}{115856201} a^{3} - \frac{1792523287720805559473}{115856201} a^{2} + \frac{6039452497813815714645}{115856201} a + \frac{524452446825637320235}{115856201} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( -a^{4} + a^{3} + 4 a^{2} - 2 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 17 a^{4} - 4 a^{3} - 81 a^{2} - 5 a + 50\) , \( -25 a^{4} - 51 a^{3} + 176 a^{2} + 178 a - 273\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-2a-1\right){x}^{2}+\left(17a^{4}-4a^{3}-81a^{2}-5a+50\right){x}-25a^{4}-51a^{3}+176a^{2}+178a-273$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.